Finite Group Databasebeta

← C₃×C₆₇×D₆₇ #41002233

Subgroups — C₃×C₆₇×D₆₇ #41002233

Order 26934.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₆₇²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₆₇×D₆₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₆₇²	C₂×C₃	1	1	yes	characteristic, complemented, Sylow
4	C₂×C₃×C₆₇	—	67	1	no	maximal
5	C₃×D₆₇	C₆₇	1	1	yes	maximal, characteristic, direct product factor
6	C₃×C₆₇	—	2	33	no	—
7	C₃×C₆₇	D₆₇	1	1	yes	characteristic, direct product factor
8	C₃×C₆₇	C₂×C₆₇	1	1	yes	characteristic, complemented
9	C₂×C₆₇	—	67	1	no	—
10	D₆₇	C₃×C₆₇	1	1	yes	characteristic, direct product factor
11	C₆₇	—	2	33	no	—
12	C₆₇	C₃×D₆₇	1	1	yes	characteristic, direct product factor
13	C₆₇	C₂×C₃×C₆₇	1	1	yes	characteristic, complemented
14	C₂×C₃	—	67	1	no	Hall
15	C₃	C₆₇×D₆₇	1	1	yes	characteristic, direct product factor, Sylow
16	C₂	—	67	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₆₇	D₆₇
commutator, cosocle, hypocenter	C₆₇	C₂×C₃×C₆₇
Frattini	C₁	G
Fitting, socle	C₃×C₆₇²	C₂
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₃×C₆₇

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×C₆₇²	C₂	semidirect
2. C₆₇×D₆₇	C₃	direct (N × H)
3. C₆₇²	C₂×C₃	semidirect
5. C₃×D₆₇	C₆₇	direct (N × H)
7. C₃×C₆₇	D₆₇	direct (N × H)
8. C₃×C₆₇	C₂×C₆₇	semidirect
10. D₆₇	C₃×C₆₇	direct (N × H)
12. C₆₇	C₃×D₆₇	direct (N × H)
13. C₆₇	C₂×C₃×C₆₇	semidirect
15. C₃	C₆₇×D₆₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₆₇²	2
C₆₇×D₆₇	3
C₂×C₃×C₆₇	67
C₃×D₆₇	67